Functional Analysis
Operator K-theory
Цели урока
- Understand the construction of K_0(A) via Murray-von Neumann classes of projections
- Master K_1(A) via unitary homotopies and the six-term exact sequence
- Know Bott periodicity and its consequence K_{n+2} = K_n
- Understand the Atiyah-Singer theorem as a pairing of K-theoretic classes
Предварительные знания
- C*-algebras and von Neumann algebras
- Banach algebras
- Topology and basic K-theory
Why are topological phases of matter, insulators, superconductors, the quantum Hall effect, stable under perturbations? Because they are classified by integers in the K_0 group, and integers cannot change continuously.
- Topological insulators (Nobel Prize 2016): the Chern number C_1 in K_0 classifies the topological phase. Bi_2Se_3 has C_1 = 1
- Quantum Hall effect: conductivity is exactly e^2/h * integer, a Chern number, a K-theoretic invariant
- Quantum computing: protected qubits built on topological insulators use K_0 invariants for resilience to decoherence
- Neural networks and topological features: Chern numbers of neural data fields are a research direction in topological ML
From topological K-theory to operator K-theory
In 1957 Grothendieck created the K-theory of algebraic bundles over schemes (algebraic K-theory). Atiyah and Hirzebruch in 1961 defined topological K-theory K^0(X) as the Grothendieck group of bundles on X. Atiyah and Singer in 1963 to 1968 proved the index theorem. Brown, Douglas, and Fillmore in 1973 to 1977 linked K-theory and C*-algebra extensions (BDF theory). Kasparov in 1980 to 1981 built KK-theory and generalized everything. In 2016 the Nobel committee noted: topological phases of matter are K-theory in action.
K_0 as the group of projection classes
The 2016 Nobel Prize in Physics recognized topological phases of matter, including topological insulators. Topological phases are classified by Chern numbers, elements of the K_0 group of the C*-algebra of the Hamiltonian. Atiyah and Hirzebruch defined topological K-theory of spaces in 1961. Kasparov extended it to C*-algebras in the 1980s through KK-theory.
The Chern number is an element of K_0 of the C*-algebra of a bulk quantum Hamiltonian. For topological insulators it takes only integer values (Chern quantization), and that is exactly what makes topological phases stable to small perturbations.
What does K_0(A) classify for a C*-algebra A?
K_0(A) = Gr(V(A)), the Grothendieck group of projection classes. Two projections p, q in matrix algebras over A are equivalent (p ~ q) iff a partial isometry v exists with v*v = p, vv* = q.
K_1 and the six-term exact sequence
K_1(A) is the group of unitary elements of the algebra modulo homotopy. For C(S^1): K_0 = Z x Z, K_1 = Z x Z. The winding number of a loop in GL_n is an element of K_1. The six-term exact sequence, the K-theoretic analog of the Mayer-Vietoris sequence in topology, ties K_0 and K_1 of an algebra and its ideal together.
| Algebra A | K_0(A) | K_1(A) | Interpretation |
|---|---|---|---|
| C | Z | 0 | Rank of a projection (natural number) |
| M_n(C) | Z | 0 | Rank of a projection in M_n |
| K(H) (compact operators) | Z | 0 | Countable rank of a compact projection |
| C(S^1) functions on the circle | Z | Z | K_0: [1], K_1: winding number |
| C(S^2) | Z x Z | 0 | K_0: rank + Chern number |
| C_0(R) | 0 | Z | Periodicity: K_1(C_0(R)) = K_0(C) = Z |
What does Bott periodicity say in K-theory?
Bott periodicity: K_0(SA) = K_1(A), K_1(SA) = K_0(A). Corollary: K_n(A) = K_{n mod 2}(A). Computing K_0 and K_1 is enough; the rest repeats.
Kasparov KK-theory and applications
Kasparov built KK-theory in 1980-1981: the bifunctor KK(A, B) generalizes K-theory and C*-algebra extensions. The Kasparov intersection product KK(A, B) x KK(B, C) -> KK(A, C) is the noncommutative analog of the Atiyah-Singer theorem. Applications: proofs of the Novikov conjecture for wide classes of groups.
Topological insulators and K-theory
K_0 as a classifying space
A topological insulator is a quantum material with a bulk gap and conducting edge states. The phase is classified by the Chern number C_1 in K_0 of the Hamiltonian's C*-algebra. A real example: Bi_2Se_3 has C_1 = 1, a nontrivial topological phase. This is topologically protected: small perturbations cannot move C_1 = 1 to C_1 = 0 without closing the gap.
What does bivariant K-theory KK(A, B) compute?
KK(A, B) is the set of homotopy classes of Kasparov (A, B)-bivariant modules. Kasparov intersection product: KK(A, B) x KK(B, C) -> KK(A, C), the noncommutative theorem on composed indices.
Index theorems via K-theory
The Atiyah-Singer theorem (1963) is one of the central results of twentieth-century mathematics. The index dim ker D - dim ker D* of an elliptic operator D on a compact manifold M is computed through topological characteristic classes. An analytic number equals a topological number, a deep bridge between analysis and topology.
Special cases of Atiyah-Singer: Gauss-Bonnet (ind = Euler number chi(M)), Hirzebruch signature theorem (ind = signature of the intersection form), Riemann-Roch (ind = dim H^0 - dim H^1 for divisors). All follow from one formula.
What does the Atiyah-Singer index theorem say?
Atiyah-Singer theorem: ind(D) = dim ker D - dim ker D* = integral of A_hat(M) wedge ch(sigma(D)). The index is an integer-valued topological invariant, stable under compact perturbations of D.
Links to other topics
Operator K-theory is a bridge between algebra, topology, and physics.
- C*-algebras and von Neumann algebras — Operator K-theory is a functor from C*-algebras to abelian groups
- Connes' noncommutative geometry — Pairing K-theory with cyclic cohomology gives the noncommutative Atiyah-Singer index
- Banach algebras — K-theory transfers to Banach algebras through idempotents and invertibles
Итоги
- K_0(A): Grothendieck group of projection classes p ~ q (Murray-von Neumann). K_0(M_n) = Z
- K_1(A): homotopy classes of unitary matrices GL_n(A) / GL_n(A)_0
- Six-term exact sequence for an ideal I in A. Boundary map partial
- Bott periodicity: K_0(SA) = K_1(A), K_1(SA) = K_0(A). K-theory is 2-periodic
- Atiyah-Singer theorem: ind(D) = integral of A_hat(M) wedge ch(sigma(D)). Analytic = topological
Вопросы для размышления
- Why do K-theoretic invariants (Chern numbers) take only integer values, and how does that tie to the stability of topological phases?
- What is the deep connection between K_0 (projections) and K_1 (unitaries) through the exact sequence and Bott periodicity?
- How does the Atiyah-Singer theorem unify Gauss-Bonnet, Riemann-Roch, and the signature theorem under one formula?